, Volume 50, Issue 1, pp 46-61
Date: 02 Apr 2011

Growth in Poisson algebras

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A criterion for polynomial growth of varieties of Poisson algebras is stated in terms of Young diagrams for fields of characteristic zero. We construct a variety of Poisson algebras with almost polynomial growth. It is proved that for the case of a ground field of arbitrary characteristic other than two, there are no varieties of Poisson algebras whose growth would be intermediate between polynomial and exponential. Let V be a variety of Poisson algebras over an arbitrary field whose ideal of identities contains identities {{x 1, y 1}, {x 2, y 2}, . . . , {x m , y m }} = 0 and {x 1, y 1} · {x 2, y 2} · . . . · {x m , y m } = 0, for some m. It is shown that the exponent of V exists and is an integer. For the case of a ground field of characteristic zero, we give growth estimates for multilinear spaces of a special form in varieties of Poisson algebras. Also equivalent conditions are specified for such spaces to have polynomial growth.

Translated from Algebra i Logika, Vol. 50, No. 1, pp. 68–88, January–February, 2010.